Naturality of Quasi-Invariance of Some Measures

The purpose of this paper is to show by examples that, in a noncommutative setting, many natural measures do not enjoy the quasi-invariance à la Cameron—Martin which comes immediately to mind. Noncommutativity means the nontriviality of the bracket of the natural vector fields involved. In finite dimension brackets of the natural vector fields associated with an elliptic operator do not increase the tangent space, which is, by the ellipticity assumption, generated already by those vector fields. In infinite dimension the phenomena is quite different: starting with an elliptic operator (for instance the Laplacian on a Riemann Hilbert manifold M), the bracket phenomena will produce immediately a new tangent space, which could not necessarily contain the given tangent space of M. A well-known fact is that measures associated to the Brownian motion on M have to be realized as a Borelian measure on a bigger space than M. In the same way, a new tangent space to M has to be realized.